CHAPTER 1
Models for Integrated-Circuit Active Devices

1.1 Introduction

The analysis and design of integrated circuits depend heavily on the utilization of suitable models for integrated-circuit components. This is true in hand analysis, where fairly simple models are generally used, and in computer analysis, where more complex models are encountered. Since any analysis is only as accurate as the model used, it is essential that the circuit designer have a thorough understanding of the origin of the models commonly utilized and the degree of approximation involved in each.

This chapter deals with the derivation of large-signal and small-signal models for integrated-circuit devices. The treatment begins with a consideration of the properties of junctions, which are basic parts of most integrated-circuit elements. Since this book is primarily concerned with circuit analysis and design, no attempt has been made to produce a comprehensive treatment of semiconductor physics. The emphasis is on summarizing the basic aspects of semiconductor-device behavior and indicating how these can be modeled by equivalent circuits.

1.2 Depletion Region of a pn Junction

The properties of reverse-biased junctions have an important influence on the characteristics of many integrated-circuit components. For example, reverse-biased junctions exist between many integrated-circuit elements and the underlying substrate, and these junctions all contribute voltage-dependent parasitic capacitances. In addition, a number of important characteristics of active devices, such as breakdown voltage and output resistance, depend directly on the properties of the depletion region of a reverse-biased junction. Finally, the basic operation of the junction field-effect transistor is controlled by the width of the depletion region of a junction. Because of its importance and application to many different problems, an analysis of the depletion region of a reverse-biased junction is considered below. The properties of forward-biased junctions are treated in Section 1.3 when bipolar-transistor operation is described.

Consider a junction under reverse bias as shown in Fig. 1.1. Assume constant doping densities of atoms/cm3 in the -type material and atoms/cm3 in the -type material. (The characteristics of junctions with nonconstant doping densities will be described later.) Due to the difference in carrier concentrations in the -type and -type regions, there exists a region at the junction where the mobile holes and electrons have been removed, leaving the fixed acceptor and donor ions. Each acceptor atom carries a negative charge and each donor atom carries a positive charge, so that the region near the junction is one of significant space charge and resulting high electric field. This is called the depletion region or space-charge region. It is assumed that the edges of the depletion region are sharply defined, as shown in Fig. 1.1, and this is a good approximation in most cases.

Figure 1.1 The abrupt junction under reverse bias . (a) Schematic. (b) Charge density. (c) Electric field. (d ) Electrostatic potential.

For zero applied bias, there exists a voltage across the junction called the built-in potential. This potential opposes the diffusion of mobile holes and electrons across the junction in equilibrium and has a value1

where

the quantity is the intrinsic carrier concentration in a pure sample of the semiconductor and at K for silicon.

In Fig. 1.1, the built-in potential is augmented by the applied reverse bias, , and the total voltage across the junction is . If the depletion region penetrates a distance into the -type region and into the -type region, then we require

because the total charge per unit area on either side of the junction must be equal in magnitude but opposite in sign.

Poisson's equation in one dimension requires that

where is the charge density, is the electron charge ( coulomb), and is the permittivity of the silicon ( farad/cm). The permittivity is often expressed as

where is the dielectric constant of silicon and is the permittivity of free space ( F/cm). Integration of (1.3) gives

where is a constant. However, the electric field is given by

Since there is zero electric field outside the depletion region, a boundary condition is

and use of this condition in (1.6) gives

Thus the dipole of charge existing at the junction gives rise to an electric field that varies linearly with distance.

Integration of (1.7) gives

If the zero for potential is arbitrarily taken to be the potential of the neutral -type region, then a second boundary condition is

and use of this in (1.8) gives

At , we define , and then (1.9) gives

If the potential difference from to is , then it follows that

and thus the total voltage across the junction is

Substitution of (1.2) in (1.12) gives

From (1.13), the penetration of the depletion layer into the -type region is

Similarly,

Equations 1.14 and 1.15 show that the depletion regions extend into the -type and -type regions in inverse relation to the impurity concentrations and in proportion to . If either or is much larger than the other, the depletion region exists almost entirely in the lightly doped region.

EXAMPLE

An abrupt junction in silicon has doping densities atoms/cm3 and atoms/cm3. Calculate the junction built-in potential, the depletion-layer depths, and the maximum field with 10 V reverse bias.

From (1.1),

From (1.14), the depletion-layer depth in the -type region is

The depletion-layer depth in the more heavily doped -type region is

Finally, from (1.7), the maximum field that occurs for is

Note the large magnitude of this electric field.

1.2.1 Depletion-Region Capacitance

Since there is a voltage-dependent charge associated with the depletion region, we can calculate a small-signal capacitance as follows:

Now

where is the cross-sectional area of the junction. Differentiation of (1.14) gives

Use of (1.17) and (1.18) in (1.16) gives

The above equation was derived for the case of reverse bias applied to the diode. However, it is valid for positive bias voltages as long as the forward current flow is small. Thus, if represents the bias on the junction (positive for forward bias, negative for reverse bias), then (1.19) can be written as

where is the value of for .

Figure 1.2 Charge density versus distance in a graded junction.

Figure 1.3 Behavior of junction depletion-layer capacitance as a function of bias voltage .

Equations 1.20 and 1.21 were derived using the assumption of constant doping in the -type and n-type regions. However, many practical diffused junctions more closely approach a graded doping profile, as shown in Fig. 1.2. In this case, a similar calculation yields

Note that both (1.21) and (1.22) predict values of approaching infinity as approaches . However, the current flow in the diode is then appreciable and the equations are no longer valid. A more exact analysis[2, 3] of the behavior of as a function of gives the result shown in Fig. 1.3. For forward-bias voltages up to about , the values of predicted by (1.21) are very close to the more accurate value. As an approximation, some computer programs approximate for by a linear extrapolation of (1.21) or (1.22).

EXAMPLE

If the zero-bias capacitance of a diffused junction is 3 pF and V, calculate the capacitance with 10 V reverse bias. Assume the doping profile can be approximated by an abrupt junction.

From (1.21),

1.2.2 Junction Breakdown

From Fig. 1.1c, it can be seen that the maximum electric field in the depletion region occurs at the junction; and for an abrupt junction, (1.7) yields a value

Substitution of (1.14) in (1.23) gives

where has been neglected. Equation 1.24 shows that the maximum field increases as the...